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arXiv 2106.08705cond-mat.stat-mechmath.PR
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通过随机介质中定向聚合物的大偏差抽样观测静态Kardar-Parisi-Zhang界面的最优路径

Observing symmetry-broken optimal paths of stationary Kardar-Parisi-Zhang interface via a large-deviation sampling of directed polymers in random media

Alexander K. Hartmann, Baruch Meerson, Pavel Sasorov

备注：
11 pages, 9 figures
发表信息：
Phys. Rev. E 104, 054125 (2021)
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摘要：

考虑由Kardar-Parisi-Zhang方程描述的固定界面$h(x,τ)$的一点界面高度差$h(x=0,τ=t)-h(x=0,τ=0)=H$的短时概率分布$\mathcal{P}(H,t)$。先前已经证明，最佳路径--主导$\mathcal{P}(H,t)$上尾的界面$h(x,τ)$的最可能的历史--由$h(x,τ)$向左或向右行进的任何\emph{two}斜坡状结构来描述。这两个解决方案出现，在$H$的临界值，通过自发破缺的镜像对称性$x\leftrightarrow -x$的最佳路径，这种对称性破缺是负责在系统中的二阶动力学相变。我们采用大偏差蒙特卡罗抽样算法结合KPZ界面和定向聚合物在高温下的随机势之间的映射，数值模拟的接口配置。这使我们能够观察到最佳路径，它决定了$\mathcal{P}(H,t)$的两个尾部中的每一个，直到概率密度小到$10^{-500}$。在很短的时间内，我们观察到镜像对称破坏旅行的最佳路径上尾，和一个单一的镜像对称路径下尾，在良好的定量分析预测。在很长一段时间，即使在中等值的$H$，最佳波动方法是\emph{not}应该适用，我们仍然观察到两个明确的主导路径。它们中的每一个都违反了镜像对称性$x\leftrightarrow -x$，并且是另一个的镜像。

英文摘要：

Consider the short-time probability distribution $\mathcal{P}(H,t)$ of the one-point interface height difference $h(x=0,τ=t)-h(x=0,τ=0)=H$ of the stationary interface $h(x,τ)$ described by the Kardar-Parisi-Zhang equation. It was previously shown that the optimal path -- the most probable history of the interface $h(x,τ)$ which dominates the upper tail of $\mathcal{P}(H,t)$ -- is described by any of \emph{two} ramp-like structures of $h(x,τ)$ traveling either to the left, or to the right. These two solutions emerge, at a critical value of $H$, via a spontaneous breaking of the mirror symmetry $x\leftrightarrow -x$ of the optimal path, and this symmetry breaking is responsible for a second-order dynamical phase transition in the system. We simulate the interface configurations numerically by employing a large-deviation Monte Carlo sampling algorithm in conjunction with the mapping between the KPZ interface and the directed polymer in a random potential at high temperature. This allows us to observe the optimal paths, which determine each of the two tails of $\mathcal{P}(H,t)$, down to probability densities as small as $10^{-500}$. At short times we observe mirror-symmetry-broken traveling optimal paths for the upper tail, and a single mirror-symmetric path for the lower tail, in good quantitative agreement with analytical predictions. At long times, even at moderate values of $H$, where the optimal fluctuation method is \emph{not} supposed to apply, we still observe two well-defined dominating paths. Each of them violates the mirror symmetry $x\leftrightarrow -x$ and is a mirror image of the other.

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